- 55 Views
- Uploaded on
- Presentation posted in: General

2145-391 Aerospace Engineering Laboratory I

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

- Measurement
- Measurement Errors / Models
- Measurement Problem and The Corresponding Measurement Model
- Measure with Single Instrument:Single Sample / Multiple Samples
- Measure with Multiple Instruments:Single Sample / Multiple Samples

- Measurement Problem and The Corresponding Measurement Model
- Uncertainty of A Measured Quantity (VS Uncertainty of A Derived Quantity – Next Week)
- Measurement Statement
- Single Sample Measurement
- Multiple Samples Measurement

- Where are we?
- Measured Quantity VSDerived Quantity
- Objectives and Motivation
- Deterministic Phenomena VS Random Phenomena
- Measurement Problems, Measurement Errors, Measurement Models
- Population and Probability
- Probability Distribution Function (PDF)
- Probability Density Function (pdf)
- Expected Value
- Moments

- Sample and Statistics
- Sample Mean and Sample Variance

- Interval Estimation
- Terminologies for Measurement: Bias, Precise, Accurate
- Error VS Uncertainty
- Measurement Statement
- Measured Variable as A Random Variable
- Uncertainty of A Measured Quantity [VS Uncertainty of A Derived Quantity – Next Week]
- Measurement Statement

- Experimental Program: Test VS Sample
- Some Uses of Uncertainty

- Week 1: Knowledge and Logic
- Week 2:
- Structure and Definition of an Experiment
- DRD/DRE

bottommost level

bottommost level

Week 3: Instruments

Week 4: Measurement and Measurement Statement:

Recall the difference between

Measured Quantityy

(numerical value of yis determined

from measurement with an instrument)

VS

Derived Quantityy

(numerical value of y is determined

from a functional relation)

In this period we focus first on measured quantity.

Measurement Statement:

If you

- know what this measurement statement says (regarding measurement result),
- know its use [what good is if for, and when to use it]
- know and understand its underlying ideas [why should we report a measurement result with this statement],
- (know, understand, and) know how to report a measurement result of a measured quantity with this measurement statement for the cases of
- A single-sample measurement
- A multiple-sample measurement
we can all go home.

Activity: Class Discussion on the above

Right. I can just simply close my eyes and bet against

- 10 students come up and measure the resistance of a given resistor.
- Then, report the measurement result.
Discussions

- Do they get the same result?
- If not, what is, and who gets, the ‘correct’ value then?

Wanna bet500 bahts? [on whom, or which value]

- If the next 10 of your friends come up and measure the resistance, and
less than 9 out of 10 of themdo not get the value you bet on,

you lose and give me 500 bahts. Otherwise, I win.

Or

- I’ll let you set the term of our bet.[Of course, I have to agree on the term first.]
What term should our bet be then? [Make it reasonable and “bettable.”]

Given that there are some (random) variations in repeated measurements,

how should we report a measurement result so that it makes some usable sense?

The Why

The Use

The How

of Measurement Statement

Deterministic Phenomena VS Random Phenomena

Deterministic Variable VS Random Variable

In reality, however, chances are that we will not have that exact position

Deterministic Phenomena

(1) The state of the system at time t, and

(2) its governing relation,

deterministically determine the state of the system – or the value of the deterministic variable y -

at any later time.

Example: Free fall (and Newton’s Second law)

A random or statistical experiment is an experiment in which [1]

- all outcomes of the experiment are known in advance,
- any realization (or trial) of the experiment results in an outcome that is not known in advance, and
- the experiment can be repeated under nominally identical condition.
[1]Rohatgi, V. K., 1976, An Introduction to Probability Theory and Mathematical Statistics, Wiley, New York, p. 20.

Class Activity:Discussion

Is tossing a coin a random experiment?

- All possible outcomes are H and T and nothing more.
- For any toss, we cannot know/predict the outcome in advance.
HorT?

3.Care can be taken to repeat the toss under nominally identical condition.

Measurement:

- All possible outcomes are known in advance, e.g.,
- For any one realization w, we cannot know/predict the exact outcome with certainty in advance.
- Care can be taken to repeat the measurement under nominally identical condition.

Class Activity:Discussion

Is measurement a random experiment?

Measurement Problems

Measurement Errors

Measurement Models

- Measure with a single instrument
- Single Sample
- Multiple Samples

- Single Sample
- Multiple Samples

X

Measured value of X at reading/sample i :

True value ofX :

Total measurement error for reading i :

Error at measurement i

Measurement i

We never know the true value.

X

Systematic/Bias error:

Constant. Does not change with realization.

Random/Precision error for reading i,

Randomly varying from one realization to another.

Measurement model

Statistical Experiment:

Repeated measurements under nominally identical condition

Measurement i

Measurement Model 1:

The ith measured value

The ith observed value of a random variableX

True value

[Can never be known for certainty, not a random variable]

Systematic / Bias error

[Constant. Does not change with realization,not a random variable]

Random / Precision error

[Randomly varying from one realization to another, a random variable]

Frequency of occurrences

the population distribution

of measured valueX

X

Statistical Experiment:

Repeated measurements under nominally identical condition

Measurement i

Repeated measurements under nominally identical condition

Description of Deterministic Variable:

State the numerical value of the variable under that condition.

Description of Random Variable

Since it randomly varies from one realization to the next

[even under the same nominal condition and we cannot predict its value exactly in advance],

a meaningful way to describe it is by describing its probability (distribution).

VS

Population

Sample

Probability – Deductive reasoning

Given properties of population,

extract information regarding a sample.

Statistics – Inductive reasoning

Given properties of a sample,

extract information regarding the population.

Population

Sample

Population and Probability

Probability – Deductive Reasoning

Given properties of population,

extract information regarding a sample.

Some Properties

x

The probability of an event is the value of

Some Properties

Area

or

= Area under the pdf curve from to x.

x

Thin / high VS Wide / low

x

PDF FX(x)

pdf fX(x)

x

- The probability of an event is
- 1. the area under the fX(x) curve from to x,
- 2. the value of FX(x).

Area

x2

x1

FX(x2)

PDF FX(x)

pdf fX(x)

FX(x2) - FX(x1)

FX(x1)

x

- The probability of an event is
- 1. the area under the fX(x) curve from x1 to x2,
- 2. the value of [FX(x2) - FX(x1)].

PDF: U(x;-1,1)

PDF: U(x;-2,2)

U(x;-1,1)

U(x;-2,2)

x

Function U(x) with parameters a and b:

Thin / high VS Wide / low

x

Function N(x) with parameters m and s 2:

PDF: N(x;0,1)

PDF: N(x;0,2)

N(x;0,1)

N(x;0,2)

x

Function t(x) with parameter n :

PDF: t(x; 20)

PDF: t(x; 5)

t(x; 5)

t(x; 20)

x

Function c2(x) with parameter n :

c 2(x; 5)

c 2(x; 10)

c 2(x; 15)

c 2(x; 20)

Definition: Expected Value of A Random Variable X

The expected value (or the mathematical expectation or the statistical average) of a continuous random variable X with a pdf fX(x) is defined as

Definition:Expected Value of A Function of A Random Variable X

Let Y = g(X) be a function of a random variable X, then Yis also a random variable,

and we have

However, we can also calculate E(Y) from the knowledge of fX(x) without having to refer to fY(y)as

Definition:Moment About The Origin

The rth-ordermoment about the origin (of a df) of X, if it exists, is defined as

where r = 0,1,2,….

Note that this is the rth-order moment of area under fX(x)about the origin.

Definition:Central Moment

The rth-order central momentof a df of X, if it exists, is defined as

where r = 0,1,2,….and E[X] = mX.

Note that this is the rth-order moment of area under fX(x)about mX.

fX (x)

x

dx

dA

dA

moment arm for Mr = x(r)

moment arm for mr = (x-mX)(r)

Area dA = fX(x)dx

NOTE: Due to the rth-power of the arm length, the values of fX (x) at further distance from the center (origin or mX) relatively contribute more to the moment than those at closer to the center.

x

x - mx

mX

Properties of Origin Moment Mr

Moment order 0:

Moment order 1 (Mean of rv X):

Moment order 2

Properties of Central Moment mr

Moment order 0:

Moment order 1:

Moment order 2:

(Variance sX2)

mXis the location of the centroid of the pdf.

sX2is a measure of the width of the pdf.

Population

Sample

Population Mean

Population Variance

Sample Mean

Sample Variance

How close is to in some sense?

Sample and Statistics

Since we do not know the properties of the population ,

we want to estimate them with the statistics drawn from a sample.

Statistics – Inductive reasoning

Definition

Let X1, X2, …, Xnbe a random sample from a distribution function fX(x).

Then, the following statistics are defined.

Sample Mean:

Sample Variance:

Sample Standard Deviation:

Sample mean, sample variance, and sample standard deviation are statistics, hence, random variables, not simply numbers.

Unbiased estimator of mX.

Unbiased estimator of .

Interval Estimation

Assume X is a random variable whose pdf is normal and

Let (X1, X2, …, Xn)be an iid random sample from

- Interval Estimation: Probability Distributions of Random Variables

Normal and Students t

Chi Squared

Area = a/2

Area = a/2

Theorem 1: Standard Normal Random Variable

If , then .

Zis called a standard normal random variable.

In addition, we have

or

whereza/2 denotes the value on the z axis for which a/2of the area under the z curve lies to the right of za/2.

magnitude of the deviation/distance

from X to mX, or from mX to X.

The probability that deviates from no more than ( times ) is

.

Theorem 2:Distribution for A Random Variable

Let (X1, X2, …, Xn)be a sample from .

Then, the random variable has

Hence,

or

The probability that deviates from no more than ( times ) is .

Theorem 3:Distribution for A Random Variable

(Student’s t Distribution)

Let (X1, X2, …, Xn)be a sample from .

Then, the random variable has

that is, T has a Student’s t distribution with degree of freedom n = n -1.

Hence,

or

The probability that deviates from no more than ( times ) is .

One More Sample from Previously Drawn n Samples (Large Sample Size Approximate, n large)

Let (X1, X2, …, Xn)be a sample from and

be the sample variance of this sample.

Let be an additional single sample drawn from .

Then, the random variable has

that is, T has a Student-t distribution withdegree of freedom n = n-1.

Hence,

or

The probability that deviates from no more than ( times ) is .

Students t

Area = a/ 2

Interval Estimation

Assume X is a random variable whose pdf is normal and

Let (X1, X2, …, Xn)be an iid random sample from

- Interval Estimation: Probability Distributions of Random Variables

Terminologies for Measurement

Bias

Precise

Accurate

X

X

XTrue , m X

XTrue, m X

XTrue,

m X

X

X

XTrue

m X

Frequency of occurrences

Biased + Imprecise Inaccurate

Biased + Precise Inaccurate

Unbiased + Imprecise Inaccurate

Unbiased + Precise Accurate

Error VS Uncertainty

- Error
If the error is known for certainty, (it is the duty of the experimenter to) correct it and it is no longer an error.

- Uncertainty
For error that is not known for certainty, no correction scheme is possible to correct out these errors.

In this respect, the termuncertainty is more suitable.

- The two terms sometimes – if not often – are used without strictly adhere to this. Nonetheless, the above should be recognized.

Measurement Statement

Measured Variable as A Random Variable

Uncertainty of A Measured Quantity

[VS Uncertainty of A Derived Quantity – Next Week]

Measurement Statement

Sample width

Population width

ei

Area = 1- a

Bias

i

For repeated measurements under nominally identical condition:

- Bias:results in the deviation of the (population) mean from the true value.
- Precision/Random:results in the scatter in datain a set of repeated measurements.
It is viewed and quantified as

the band width of the scatter

not absolute position.

One realization of random error

+BX: Bias Uncertainty

e i

bj: One realization of bias error.

XTrue

+PX:Random Uncertainty

i

If the measurement instrument identity is changed, the bias is changed and is considered a random variable.

Xj,i=ithrealization of instrumentj

1

Measurement statementi

Bias and Precision/Random uncertainties are combined with

root-sum-square (rss) method.

- Bias Uncertainty
Bias Uncertainty = Root-sum-square (RSS) of elemental error sources.

To the very least, it is the uncertainty of the instrument itself, e.g.,

- Precision Uncertainty
Report:

Single value:

Average value:

Measurement Statement as An Interval Estimate

Experimental Program:

Test VS Sample

Test:The word is associated with the evaluation of DRD/DRE, or r.

One Test = One Evaluation of DRE One r

Measurement, Reading, Sample: The word is associated with Xireading from the individual measurement system i.

One Sample = One Measurement of Xi One Xi

ST/SS

Single Test/Single Sample

Single

Single X Single r

Single r

Single r

ST/MS

Single Test/Multiple Sample

Single r

Multiple

Average X Single r

Single r

MT/SS

Multiple Test/Single Sample

Single

Average r

Single X Single r

Multiple r

MT/MS

Multiple Test/Multiple Sample

Multiple

Average X Single r

Multiple r

Average r

Average overrk

MT/SS

Average over(Xi)k

Average overrl

MT/MS

Reading/Sample kth

Test kth

ST/SS

ST/SS

ST/MS

ST/MS

Single Sample: Report

Multiple Samples: Report

Single Sample: Report

Multiple Samples: Report

Students t

Area = a/ 2

Measurement Statement

Some Uses of Uncertainty

- Interpretation of Experimental Result
- Comparing Theory and Experiment
- Comparing Two Models
- Industry

y

y

y

x

x

x

With uncertainty, at least we have some indications of how good – or precise – is the current experimental result.

Without uncertainty, we cannot evaluate how good – or precise - is the current experimental result.

Model A

Model A

y

y

y

Model A

x

x

x

With uncertainty, in this case we see that they are consistent (within the limit of the uncertainty of the current experimental result).

Without uncertainty, we cannot compare whether or not the theoretical result is consistent with the experiment result.

- Note:
- Often a theoretical result requires constants that must – or should - be determined from experiment.
- As a result, there are uncertainty associated with a theoretical result also.
- Therefore, there should be error bars (though not shown here) associated with theoretical result also.

With uncertainty, in this case we see that they are not consistent (within the limit of the uncertainty of the current experimental result).

y

y

Model A

Model A

Model B

Model B

x

x

Without uncertainty of the experimental result,

we cannot differentiate which one, A or B, is better.

With uncertainty of the experimental result,

in this case we see that the performance of models A and B cannot be differentiated with the current experimental result.

- Would you like to know – roughly:
- How accurate (or uncertain) are the flowmeters at gas stations in Bangkok?
- Let’s say, you pay ~ 1,000 bahts/week 52,000 bahts / year.
- Is it + 10 % @ 95% CL,
- + 5 % @ 95% CL,
- + 1 % @ 95% CL,
- + 0.5 % @ 95% CL?
- Imagine, e.g., PTT who sells petrol/gas in billions of bahts.
- Since we cannot avoid this uncertainty – but we can try to minimize it, what would you do if you were, e.g., PTT, and you were uncertain ~ + 10%, + 5%, + 1%, + 0.5% ?